Cake: Adjust the oven rack to the lower-middle position and heat the oven to 325 degrees. Grease two 9-inch round cake pans with shortening and dust with flour.

Beat the egg whites and whole egg in a large measuring cup with a fork to combine. Add the cream of coconut, water, vanilla and coconut extracts and beat with a fork until thoroughly mixed.

Combine the flour, sugar, baking powder, and salt in the bowl of a standing mixer fitted with the paddle attachment. Mix on the lowest speed to combine, about 30 seconds. With the mixer still running on the lowest speed, add the butter 1 piece at a time, then beat until the mixture resembles coarse meal, with butter bits no larger than small peas, 2-21/2 minutes.

With the mixer still running, add 1 cup of the egg mixture. Increase the speed to medium-high and beat until light and fluffy, about 45 seconds. Then add the remaining cup of liquid in a steady stream (this should take about 15 seconds). Stop the mixer and scrape down the bowl with a rubber spatula, then beat at medium-high speed to combine, about 15 seconds. Batter will be thick. Divide the batter between the cake pans and level with an offset or rubber spatula. Bake until deep golden brown, the cakes pull away from sides of pans, and a toothpick inserted into the center of the cakes comes out clean, about 30 minutes (rotate the cakes after 20 minutes). Do not turn off the oven.

Cool in pans on wire racks for about 10 minutes, then loosen the cakes from the sides of the pans with a paring knife, invert the cakes onto racks, and place them so the top sides face up; cool to room temperature.

To toast the coconut: While the cakes are cooling, spread the shredded coconut on a rimmed baking sheet and toast in the oven, 15-20 minutes.

Buttercream: Combine the egg whites, sugar, and salt in the bowl of a standing mixer; set the bowl over a saucepan containing 11/2-inches of barely simmering water. Whisk constantly until this mixture is opaque and warm to the touch and registers about 120 degrees on an instant-read thermometer; about 2 minutes.

Transfer the bowl to the mixer and beat the whites on high speed with the whisk attachment until barely warm (about 80 degrees) and the whites are glossy and sticky; about 7 minutes. Reduce the speed to medium-high and beat in the butter 1 piece at a time. Beat in the cream of coconut and coconut and vanilla extracts. Stop the mixer and scrape the bowl. Continue to beat at medium-high speed until well-combined, about 1 minute.

To assemble cake: With a long serrated knife, cut both cakes in half horizontally so that each cake forms two layers. Assemble and frost the cake. Sprinkle the top of the cake with coconut. Press coconut into the sides of the cake with your hand, letting the excess fall back onto a baking sheet or piece of parchment paper. Cover leftover cake with plastic and refrigerate; bring to room temperature before serving.